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Cocycle Hopf Algebra Structures on Free Modified Rota–Baxter Algebras by Vertex-decorated Rooted Trees 顶点修饰的根树自由修正Rota-Baxter代数上的循环Hopf代数结构
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4240-8
Shanghua Zheng, Yukun Liu
{"title":"Cocycle Hopf Algebra Structures on Free Modified Rota–Baxter Algebras by Vertex-decorated Rooted Trees","authors":"Shanghua Zheng,&nbsp;Yukun Liu","doi":"10.1007/s10114-026-4240-8","DOIUrl":"10.1007/s10114-026-4240-8","url":null,"abstract":"<div><p>A modified Rota–Baxter operator originates from the convolution theorem for the Hilbert transformation by Tricomi. It also satisfies the modified classical Yang–Baxter equation discovered by Semenov-Tian-Shansky, and is applied to Lax equations and affine geometry of Lie groups later. In this paper, we provide an alternative explicit construction of free modified Rota–Baxter associative algebras from the perspective of combinatorial objects. First, we revisit a construction of free operated bialgebra structure on vertex-decorated rooted forests via a variant of the Hochschild 1-cocycle condition. Applying an isomorphism between two kinds of free operated algebras given by different carriers, we construct free modified Rota–Baxter algebras on vertex-decorated rooted forests. We then obtain a combinatorial coproduct on the free modified Rota–Baxter algebra by means of the universal property of free operated algebras, leading to a cocycle bialgebra structure and further a cocycle Hopf algebra structure on it.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"508 - 540"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Obstructions for Minimal Distal Actions 最小远端动作障碍
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4381-9
Enhui Shi, Hui Xu, Lizhen Zhou
{"title":"Obstructions for Minimal Distal Actions","authors":"Enhui Shi,&nbsp;Hui Xu,&nbsp;Lizhen Zhou","doi":"10.1007/s10114-026-4381-9","DOIUrl":"10.1007/s10114-026-4381-9","url":null,"abstract":"<div><p>In this paper, we mainly consider the nonexistences of minimal distal actions by some groups on compact manifolds, particularly on surfaces. Suppose that <i>X</i> is a compact manifold and Γ is a finitely generated group acting on <i>X</i>. We show in the following cases that Γ cannot act on <i>X</i> minimally and distally. (1) <i>X</i> is connected and the first Čech cohomology group <i>Ȟ</i><sup>1</sup>(<i>X</i>) with integer coefficients is nontrivial and Γ is amenable; (2) <i>X</i> is the 2-sphere <span>(mathbb{S}^{2})</span> or the real projective plane ℝℙ<sup>2</sup> and Γ contains no nonabelian free subgroup; (3) <i>X</i> is a closed surface and Γ is a lattice of SL(<i>n</i>, ℝ)(<i>n</i> ≥ 3).</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"415 - 426"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New Types of Singular Solutions of Conformal Q-curvature Equations 保形q曲率方程奇异解的新类型
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4379-3
Zongming Guo, Zhongyuan Liu, Fangshu Wan
{"title":"New Types of Singular Solutions of Conformal Q-curvature Equations","authors":"Zongming Guo,&nbsp;Zhongyuan Liu,&nbsp;Fangshu Wan","doi":"10.1007/s10114-026-4379-3","DOIUrl":"10.1007/s10114-026-4379-3","url":null,"abstract":"<div><p>We construct radial and non-radial singular solutions <i>u</i> ∈ <i>C</i><sup>4</sup>(<i>B</i>∖{0}) with a non-removable singular point <i>x</i> = 0 for the conformal Q-curvature equation </p><div><div><span>$$begin{cases}Delta^{2}u={rm{e}}^{u}quadtext{in},Bbackslash{0},int_{Bbackslash{0}}{rm{e}}^{u(x)},dx&lt;infty,end{cases}$$</span></div></div><p> where <i>B</i> = {<i>x</i> ∈ ℝ<sup>4</sup>: ∣<i>x</i>∣ &lt; 1}. More precisely, we can construct two different types of singular solutions <i>u</i> ∈ <i>C</i><sup>4</sup>(<i>B</i>∖{0}) of the equation, satisfying </p><div><div><span>$$vert xvert^{2}u(x)rightarrow - D&lt;0quadtext{uniformly},text{as},vert xvert rightarrow 0$$</span></div></div><p> for some <i>D</i><sub>0</sub> ≥ 0 and any <i>D</i> &gt; <i>D</i><sub>0</sub> ≥ 0, and satisfying </p><div><div><span>$$u(x)=o(vert xvert^{-2})quadtext{uniformly},text{as},vert xvert rightarrow 0$$</span></div></div><p> Moreover, detailed asymptotic expansions near <i>x</i> = 0 of these radial and non-radial singular solutions can be established. As an application, we can also obtain the existence of two types of solutions <span>(uin C^{4}(mathbb{R}^{4}backslashoverline{B}))</span> to the problem </p><div><div><span>$$begin{cases}Delta^{2}u={rm{e}}^{u}quadtext{in},mathbb{R}^{4}backslashoverline{B},int_{mathbb{R}^{4}backslashoverline{B}}{rm{e}}^{u(x)} dx&lt;inftyend{cases}$$</span></div></div><p> satisfying |<i>x</i>|<sup>−2</sup><i>u</i>(<i>x</i>) → − <i>D</i> &lt; 0 uniformly as |<i>x</i>| → ∞ for some D<sub>0</sub> ≥ 0 and any <i>D</i> &gt; <i>D</i><sub>0</sub>, and satisfying <i>u</i>(<i>x</i>) = <i>o</i>(|<i>x</i>|<sup>2</sup>) uniformly as |<i>x</i>| → ∞.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"481 - 507"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New Real-Variable Characterizations of Harmonic Functions with Mixed Morrey Traces and Applications 混合Morrey走线谐波函数的实变新表征及其应用
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4067-3
Bo Li, Tianjun Shen, Wenchang Sun, Chao Zhang
{"title":"New Real-Variable Characterizations of Harmonic Functions with Mixed Morrey Traces and Applications","authors":"Bo Li,&nbsp;Tianjun Shen,&nbsp;Wenchang Sun,&nbsp;Chao Zhang","doi":"10.1007/s10114-026-4067-3","DOIUrl":"10.1007/s10114-026-4067-3","url":null,"abstract":"<div><p>A well-known result of Stein–Weiss in 1971 said that a harmonic function, defined on the upper half-space, is the Poisson integral of a Lebesgue function if and only if it is also a Lebesgue function uniformly in the time variable. We show that a solution to the elliptic equation, defined on the upper half-space, is in the essentially-bounded-Morrey space of mixed type if and only if it can be represented by the Poisson integral of a mixed Morrey function, where a Liouville property is assumed. As applications, some new real-variable characterizations of the solution to the elliptic/parabolic equation related to the Neumann/Dirichlet problem are also considered via the gluing technology.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"384 - 414"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Uniform Measure Attractors for Non-autonomous Stochastic Evolution Systems 非自治随机进化系统的一致测度吸引子
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4332-5
Dingshi Li, Ran Li, Tianhao Zeng
{"title":"Uniform Measure Attractors for Non-autonomous Stochastic Evolution Systems","authors":"Dingshi Li,&nbsp;Ran Li,&nbsp;Tianhao Zeng","doi":"10.1007/s10114-026-4332-5","DOIUrl":"10.1007/s10114-026-4332-5","url":null,"abstract":"<div><p>This paper is concerned with uniform measure attractors for non-autonomous stochastic evolution systems. We first introduce the concept of uniform measure attractor and then provide a sufficient criterion for existence and uniqueness of such attractors. As an application, we prove the existence and uniqueness of uniform measure attractors for the stochastic Navier–Stokes equations with deterministic almost-periodic forcing and nonlinear diffusion terms.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"427 - 444"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multilinear Maximal Operators on Triebel–Lizorkin Spaces and Besov Spaces triiebel - lizorkin空间和Besov空间上的多线性极大算子
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4583-1
Feng Liu, Simin Liu, Shifen Wang
{"title":"Multilinear Maximal Operators on Triebel–Lizorkin Spaces and Besov Spaces","authors":"Feng Liu,&nbsp;Simin Liu,&nbsp;Shifen Wang","doi":"10.1007/s10114-026-4583-1","DOIUrl":"10.1007/s10114-026-4583-1","url":null,"abstract":"<div><p>In the present paper, the authors systematically study the mapping properties of multilinear maximal operators on the Triebel–Lizorkin spaces and Besov spaces. In the global setting, the authors provide a criterion on the boundedness and continuity of a class of multilinear operators on the Triebel–Lizorkin spaces and Besov spaces, which can be used to obtain the boundedness and continuity of the multilinear operators associated to balls, cubes and dyadic cubes, multilinear sharp maximal operator as well as multilinear operators of convolution type on the Triebel–Lizorkin spaces and Besov spaces. The corresponding results for the multilinear maximal operators associated to balls are also proved in the local setting.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"293 - 321"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Small Positive Values and Limit Theorems for Supercritical Branching Processes with Immigration in Random Environment 随机环境下具有迁移的超临界分支过程的小正值及极限定理
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4335-2
Yinxuan Zhao, Mei Zhang
{"title":"Small Positive Values and Limit Theorems for Supercritical Branching Processes with Immigration in Random Environment","authors":"Yinxuan Zhao,&nbsp;Mei Zhang","doi":"10.1007/s10114-026-4335-2","DOIUrl":"10.1007/s10114-026-4335-2","url":null,"abstract":"<div><p>Let (<i>Z</i><sub><i>n</i></sub>) be a supercritical branching process with immigration in a random environment. The small positive values and some lower deviation inequalities for <i>Z</i><sub><i>n</i></sub> are investigated. Based on these results, the central limit theorem of log <i>Z</i><sub><i>n</i></sub> and the Edgeworth expansion are obtained. The study is taken under the assumption that each individual produces 0 offspring with a positive probability.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"445 - 463"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Möbius Inversion and Duality for Summations of Stable Graphs Möbius稳定图和的反转与对偶性
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4262-2
Zhiyuan Wang, Jian Zhou
{"title":"Möbius Inversion and Duality for Summations of Stable Graphs","authors":"Zhiyuan Wang,&nbsp;Jian Zhou","doi":"10.1007/s10114-026-4262-2","DOIUrl":"10.1007/s10114-026-4262-2","url":null,"abstract":"<div><p>Using the stratifications of Deligne–Mumford moduli spaces <span>(overline{{cal{M}}}_{g,n})</span> indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus <i>g</i> with <i>n</i> external edges. By modifying the usual definition of zeta function and Möbius function of a poset, we introduce generalized (ℚ-valued) zeta function and generalized (ℚ-valued) Möbius function of the poset of stable graphs. We use them to proved a generalized Möbius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"269 - 292"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Virtual Element Method for the Elastic Transmission Eigenvalue Problem with Equal Elastic Tensors 等弹性张量弹性传输特征值问题的虚元法
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-15 DOI: 10.1007/s10114-026-4056-6
Jian Meng, Bingbing Xu, Fang Su, Xu Qian, Songhe Song
{"title":"Virtual Element Method for the Elastic Transmission Eigenvalue Problem with Equal Elastic Tensors","authors":"Jian Meng,&nbsp;Bingbing Xu,&nbsp;Fang Su,&nbsp;Xu Qian,&nbsp;Songhe Song","doi":"10.1007/s10114-026-4056-6","DOIUrl":"10.1007/s10114-026-4056-6","url":null,"abstract":"<div><p>The elastic transmission eigenvalue problem is fundamental to the qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the elastic transmission eigenvalue problem is neither self-adjoint nor elliptic. The aim of this work is to provide a systematic spectral approximation analysis for the VEM of the elastic transmission eigenvalue problem with equal elastic tensors. Considering standard assumptions on polygonal/polyhedral meshes, we prove the stability analysis of the associated VEM bilinear forms, which shall be applied to the well-defined property of the discrete solution operator. Then the correct approximation of spectrum for the proposed VEM scheme is proven. Necessitated by supporting the convergence analysis, a series of numerical examples are reported. In addition, some negative points of the current VEM scheme are considered, including the locking phenomenon and the influence of VEM stabilization parameters. Thanks to the flexibility of construction for the VEM space, the locking-free and stabilization-free VEM approaches are utilized to tackle with these negative aspects.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 2","pages":"322 - 356"},"PeriodicalIF":0.9,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Behavior of Observables in Renormalization 重整化中可观察对象的行为
IF 0.9 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2026-02-14 DOI: 10.1007/s10114-026-4322-7
Kaiyuan Cui, Fuzhou Gong
{"title":"The Behavior of Observables in Renormalization","authors":"Kaiyuan Cui,&nbsp;Fuzhou Gong","doi":"10.1007/s10114-026-4322-7","DOIUrl":"10.1007/s10114-026-4322-7","url":null,"abstract":"<div><p>In this paper, we introduce novel reference observables for the sake of establishing a scaling formula in the renormalization group (<i>RG</i>) equation. Firstly, using the transfer matrix method, we calculate the two point observables of the one dimensional (1D) Ising model without an external field under general boundary conditions. The results suggest that the two point observables decay exponentially except at the critical point. Corresponding to the <i>RG</i> procedure underlying the correlation function, we establish a similar procedure for new observables, which is consistent with the findings in physics. Secondly, from a dynamic perspective, we construct a random system via the stochastic quantization method. We calculate the new observables of this random system under the initial distribution satisfying the Dobrushin–Lanford–Ruelle (DLR) equations. Furthermore, we formulate a new renormalization scaling formula with respect to the two point observables. Finally, these results can be extended to any finite point observables, and are independent of the choice of system parameters.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"42 on","pages":"1481 - 1508"},"PeriodicalIF":0.9,"publicationDate":"2026-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148261645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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